Question

Plot trajectories of the given system. $${\mathbf{y}}^{\prime }=\left[\begin{array}{ll}-5& 3\\ -3& 1\end{array}\right]\mathbf{y}$$

Short answer

Given the general solution, compute and plot the trajectories for a range of initial conditions by varying the values of $c_1$ and $c_2$.

Step-by-step solution

Find Eigenvalues and Eigenvectors

First, let's find the eigenvalues and eigenvectors of the matrix $$A=\left[\begin{array}{cc}-5& 3\\ -3& 1\end{array}\right].$$ To find the eigenvalues, we need to determine the roots of the characteristic equation: $$det(A-\lambda I)=0,$$ which can be written as $$\left|\begin{array}{cc}-5-\lambda & 3\\ -3& 1-\lambda \end{array}\right|=(\lambda +5)(\lambda -1)-(-9)=0.$$ After simplifying the equation, we have $${\lambda }^2+4\lambda -4=0.$$ We can find the eigenvalues using the quadratic formula: $$\lambda =\frac{-b\pm \sqrt{b^2-4ac}}{2a},$$ where $a=1,b=4$, and $c=-4$. This results in the following eigenvalues: $${\lambda }_1=-2-\sqrt{12},{\lambda }_2=-2+\sqrt{12}.$$ Now we need to find the eigenvectors associated with these eigenvalues. For each eigenvalue, we solve the equation $(A-\lambda I)\mathbf{v}=0$. For ${\lambda }_1=-2-\sqrt{12}$, $$\left[\begin{array}{cc}-3-\sqrt{12}& 3\\ -3& 3-\sqrt{12}\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right],$$ which simplifies to $$v_2=\frac{-3-\sqrt{12}}{3}{v}_1.$$ So, a possible eigenvector for ${\lambda }_1$ is $${\mathbf{v}}_1=\left[\begin{array}{c}1\\ -1-\sqrt{3}\end{array}\right].$$ For ${\lambda }_2=-2+\sqrt{12}$, $$\left[\begin{array}{cc}-3+\sqrt{12}& 3\\ -3& 3+\sqrt{12}\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right],$$ which simplifies to $$v_2=\frac{-3+\sqrt{12}}{3}{v}_1.$$ So, a possible eigenvector for ${\lambda }_2$ is $$ \mathbf{v}_{2}=$$\left[\begin{array}{c}1\\ -1+\sqrt{3}\end{array}\right]$$.

Find the General Solution

Now that we have eigenvalues and eigenvectors, we can find the general solution of the given system. The general solution can be written as: $$\mathbf{y}(t)=c_1{e}^{{\lambda }_{1}t}{\mathbf{v}}_1+c_2{e}^{{\lambda }_{2}t}{\mathbf{v}}_2,$$ where $c_1$ and $c_2$ are constants. Substituting the eigenvalues and eigenvectors, the general solution becomes: $$ \mathbf{y}(t) = c_{1}e^{(-2-\sqrt{12})t}$$\left[\begin{array}{c}1\\ -1-\sqrt{3}\end{array}\right]$$ + c_{2}e^{(-2+\sqrt{12})t}$$\left[\begin{array}{c}1\\ -1+\sqrt{3}\end{array}\right]$$.

Plot Trajectories

Finally, it's time to plot the trajectories of the system using the general solution. We will not plot the trajectories directly here, but we can use any graphing tool (like Geogebra, Desmos, or Python with matplotlib) to plot the trajectories using the general solution. To plot the trajectories, just plot the general solution for different initial conditions, which means different values of $c_1$ and $c_2$.

Key concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is crucial when analyzing the behavior of linear systems, particularly systems of differential equations. An eigenvalue of a matrix is a special scalar that, when the matrix multiplies a certain non-zero vector called an eigenvector, the result is simply the eigenvector multiplied by that scalar. The relationship can be expressed as:
$$A\mathbf{v}=\lambda \mathbf{v}$$
where $A$ is a square matrix, $\mathbf{v}$ the eigenvector, and $\lambda$ the eigenvalue. To find these values for a matrix, we first solve the characteristic equation, which is obtained by setting the determinant of $A-\lambda I$ to zero, where $I$ represents the identity matrix. This equation is the key to unlocking the behavior of the system, as eigenvalues can describe the stability and dynamics of the system. Specifically, the signs and magnitudes of the eigenvalues indicate growth, decay, oscillation, or spiral behavior of trajectories in a phase portrait.

General Solution of Differential Equations

In the context of linear differential equations, the general solution represents a combination of all possible solutions. For a system of linear homogeneous differential equations as below:
$${\mathbf{y}}^{\prime }=A\mathbf{y}$$
where $A$ is a matrix and $\mathbf{y}$ is the vector of functions we want to find, the general solution can be expressed using the system's eigenvalues and corresponding eigenvectors. The solution takes the form:
$$\mathbf{y}(t)=c_1{e}^{{\lambda }_{1}t}{\mathbf{v}}_1+c_2{e}^{{\lambda }_{2}t}{\mathbf{v}}_2+\cdots +c_n{e}^{{\lambda }_{n}t}{\mathbf{v}}_n$$
Here, ${\lambda }_i$ are the eigenvalues, ${\mathbf{v}}_i$ the corresponding eigenvectors, and $c_i$ arbitrary constants determined by initial conditions. This solution encompasses all the possible trajectories the system can take depending on the initial conditions. It is through these expressions that we can visualize the entire behavior of a system over time. Importantly, the way in which solutions diverge or converge is conveyed by the exponentials, with the real parts of the eigenvalues playing the pivot role.

Characteristic Equation of a Matrix

The characteristic equation of a matrix is an algebraic expression that encapsulates the essence of linear transformations described by the matrix. It is obtained from the 'characteristic polynomial', which arises when you seek scalars $\lambda$ such that there exists a non-trivial vector $\mathbf{v}$ for the equation $A\mathbf{v}=\lambda \mathbf{v}$. The polynomial is derived by calculating the determinant of $A-\lambda I$, leading us to the characteristic equation:
$$det(A-\lambda I)=0$$
It is through the roots of this equation, which are the eigenvalues, that we can start to interpret the dynamics of the system represented by matrix $A$. In essence, the characteristic equation is a bridge from a matrix representation to the analysis of a system's behavior.

Phase Portrait

The phase portrait is a visual representation of a dynamic system's trajectories in the phase plane, where each point corresponds to a state of the system. When plotting the phase portrait for a system of differential equations, one typically employs the eigenvalues and eigenvectors to guide the drawing of trajectories. Here's how it ties with the previous concepts:

• The eigenvalues indicate the type of fixed points and the behaviors of trajectories nearby — repelling, attracting, or spiraling.
• The eigenvectors guide the direction of these trajectories.
• The general solution equation provides the precise paths taken for specific initial conditions.

Analyzing the phase portrait gives us a wealth of information about the dynamic system. For example, if all eigenvalues have negative real parts, the system's equilibrium is stable and all trajectories will eventually converge to it. Conversely, positive real parts indicate instability, and trajectories will diverge away. These insights are valuable in fields ranging from mechanical engineering to population dynamics, as they help predict long-term behavior of the systems studied.